Weibull

Fig.5 Weibull two-parameters dividing. Exemple of results for grey cast iron with heat treated surfaces [5],...

Weibull analysis Weibull distribution Weibull function Weibullmodulus... 

D. I. Gibbons and L. C. Vance, M Simulation Study of Estimators for the Parameters and Percentiles in the Two-Parameter Weibull Distribution, General Motors Research Publication No. GMR-3041, General Motors, Detroit, Mich., 1979. 

The region of unreliable reaction at use of created test-tools for visual detection of nitroxoline has been studied. The distribution of frequencies of the detection in region of unreliable reaction for system with Cu [Fe(CN)g] the best fit to the function of normal distribution, for other systems - function Weibulle. A detection limit (c J and other characteristics of test-systems are reduced in the table ... 

An eminently practical, if less physical, approach to inherent flaw-dependent fracture was proposed by Weibull (1939) in which specific characteristics of the flaws were left unspecified. Fractures activate at flaws distributed randomly throughout the body according to a Poisson point process, and the statistical mean number of active flaws n in a unit volume was assumed to increase with tensile stress a through some empirical relations such as a two-parameter power law... 

The parameter is a crack propagation velocity and n(e) is a crack activation law driven by the bulk tensile strain e and specified by the Weibull fracture theory... 

The Swedish engineer, Weibull, invented the following way of handling the statistics of strength. Fie defined the survival probability PJ.Vg) as the fraction of identical samples, each of volume Vg, which survive loading to a tensile stress a. Fie then proposed that... 

Fig. 18.3. (a) The Weibull distribution function, (b) When the modulus, m, changes, the survival probability changes os shown. 

Fig. 18.4. Survival probability plotted on "Weibull probability" axes for samples of volume Vq. This is just Fig. 1 8.3(b) plotted with axes that straighten out the lines of constant m.

Ps = survival probability of component V = volume of component o= tensile stress on component 1/q = volume of test sample (Tq = stress that, when applied to test sample, gives Ps = 1/e (= 0.37) m = Weibull modulus. 

Three statistieal distributions that are eommonly used in engineering are the Normal (see Figure 4.3(a)), Lognormal (see Figure 4.3(b)) and Weibull, both 2-and 3-parameter (see Figure 4.3(e) for a representation of the 2-parameter type). 

Weibull - Fatigue enduranee strength of metals and strength of eeramie materials. 

Figure 4.3 Shapes of the probability density function (PDF) for the (a) normal, (b) lognormal and (c) Weibull distributions with varying parameters (adapted from Carter, 1986)...

Data that is not evenly distributed is better represented by a skewed distribution such as the Lognormal or Weibull distribution. The empirically based Weibull distribution is frequently used to model engineering distributions because it is flexible (Rice, 1997). For example, the Weibull distribution can be used to replace the Normal distribution. Like the Lognormal, the 2-parameter Weibull distribution also has a zero threshold. But with increasing numbers of parameters, statistical models are more flexible as to the distributions that they may represent, and so the 3-parameter Weibull, which includes a minimum expected value, is very adaptable in modelling many types of data. A 3-parameter Lognormal is also available as discussed in Bury (1999). 

The price of flexibility comes in the difficulty of mathematical manipulation of such distributions. For example, the 3-parameter Weibull distribution is intractable mathematically except by numerical estimation when used in probabilistic calculations. However, it is still regarded as a most valuable distribution (Bompas-Smith, 1973). If an improved estimate for the mean and standard deviation of a set of data is the goal, it has been cited that determining the Weibull parameters and then converting to Normal parameters using suitable transformation equations is recommended (Mischke, 1989). Similar estimates for the mean and standard deviation can be found from any initial distribution type by using the equations given in Appendix IX. 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

The above process above could also be performed for the 3-parameter Weibull distribution to compare the correlation coefficients and determine the better fitting distributional model. Computer-based techniques have been devised as part of the approach to support businesses attempting to determine the characterizing distributions... 

It has been shown that the ultimate tensile strength, Su, for brittle materials depends upon the size of the speeimen and will deerease with inereasing dimensions, sinee the probability of having weak spots is inereased. This is termed the size effeet. This size effeet was investigated by Weibull (1951) who suggested a statistieal fune-tion, the Weibull distribution, deseribing the number and distribution of these flaws. The relationship below models the size effeet for deterministie values of Su (Timoshenko, 1966). 

As ean be seen from the above equation, for brittle materials like glass and eeramies, we ean seale the strength for a proposed design from a test speeimen analysis. In a more useful form for the 2-parameter Weibull distribution, the probability of failure is a funetion of the applied stress, L. 

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